Limit distributions and scaling functions

TL;DR

This paper reviews the asymptotic behavior of lattice polygon models, focusing on limit distributions and scaling functions, especially in the uniform perimeter ensemble, providing geometric insights into their scaling behavior.

Contribution

It offers a pedagogic review connecting limit distributions to the scaling behavior of generating functions for lattice polygons.

Findings

Limit distributions relate to the scaling of perimeter and area generating functions.

Provides geometric interpretation of scaling functions.

Summarizes known results on asymptotic behavior of lattice polygons.

Abstract

We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs with the same probability. We relate limit distributions to the scaling behaviour of the associated perimeter and area generating functions, thereby providing a geometric interpretation of scaling functions. To a major extent, this article is a pedagogic review of known results.

Tables

Table 1. Table 1: Exponents and area limit laws for prominent polygon models. An asterisk denotes a numerical analysis.

Model	$\varphi$	$\theta$	${\gamma }_0$	Area limit law
rectangles
convex polygons	$\frac{1}{2}$	$-1$	$2$	${\beta }_{1,1/2}$
Ferrers diagrams
stacks	$\frac{1}{2}$	$-\frac{1}{2}$	$1$	Gaussian
staircase polygons
bargraph polygons
column-convex polygons
directed column-convex polygons	$\frac{2}{3}$	$\frac{1}{3}$	$-\frac{1}{2}$	Airy
diagonally convex directed polygons
rooted self-avoiding polygons∗
directed convex polygons	$\frac{2}{3}$	$-\frac{1}{3}$	$\frac{1}{2}$	meander
diagonally convex polygons∗			$-\frac{1}{2}$
three-choice polygons			0

Equations

$$\mathbb{P}({\tilde{X}}_m=n)=\frac{p_{m,n}}{{\sum }_n{p}_{m,n}},\mathbb{P}({\tilde{Y}}_n=m)=\frac{p_{m,n}}{{\sum }_m{p}_{m,n}}.$$

$$\mathbb{P}({\tilde{X}}_m(q)=n)=\frac{p_{m,n}{q}^n}{{\sum }_n{p}_{m,n}{q}^n}.$$

$$\mathbb{P}({\tilde{Y}}_n(x)=m)=\frac{p_{m,n}{x}^m}{{\sum }_m{p}_{m,n}{x}^m},$$

$$P(x,q)=\sum _{m,n}{p}_{m,n}{x}^m{q}^n.$$

$$\mathbb{P}(\tilde{X}(x,q)=(m,n))=\frac{p_{m,n}{x}^m{q}^n}{{\sum }_{m,n}{p}_{m,n}{x}^m{q}^n}.$$

$$A^k{\left(x\frac{\partial }{\partial x}\right)}^{k}P(x,q)\ll {\left(q\frac{\partial }{\partial q}\right)}^{k}P(x,q)\ll B^k{\left(x\frac{\partial }{\partial x}\right)}^{2k}P(x,q),$$

$${\frac{{\partial }^k}{\partial x^k}P(x,q)|}_{x=x_c}$$

$${\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}$$

$${\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}=\underset{\begin{array}{c}q\to 1\\ -1<q<1\end{array}}{\lim }\frac{{\partial }^k}{\partial q^k}P(x,q).$$

$$g_k(x)=\frac{1}{k!}{\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}.$$

$$a_m\sim A{x}_c^{-m}{m}^{\gamma -1}(m\to \infty ),$$

$$\left(g(x)-g^{(reg)}(x)\right)\sim \frac{A\Gamma (\gamma )}{(1-x/x_c{)}^{\gamma }}(x\nearrow x_c),$$

$$g(x)\sim \frac{1}{{\left(1-x/x_c\right)}^{\gamma }}(x\to x_c\text{ in }\Delta ).$$

$$[x^m]g(x)\sim \frac{1}{\Gamma (\gamma )}{x}_c^{-m}{m}^{\gamma -1}(m\to \infty ),$$

$$[x^m]f(x)\sim \frac{1}{\Gamma (\gamma )}{x}_c^{-m}{m}^{\gamma -1}(m\to \infty ).$$

$$\frac{1}{k!}\sum _n(n{)}_k{p}_{m,n}\sim A_k{x}_c^{-m}{m}^{{\gamma }_k-1}(m\to \infty ),$$

$$g_k^{(sing)}(x)\sim \frac{f_k}{(1-x/x_c{)}^{{\gamma }_k}}(x\nearrow x_c),$$

$${\left({\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}\right)}^{(sing)}\sim {\left({{\left(q\frac{\partial }{\partial q}\right)}^{k}P(x,q)|}_{q=1}\right)}^{(sing)}(x\nearrow x_c).$$

$$\mathbb{P}({\tilde{X}}_m=n)=\frac{p_{m,n}}{{\sum }_n{p}_{m,n}}.$$

$$\begin{array}{rl}\mathbb{E}[{\tilde{X}}_m^k]& =\sum _{l=1}^{m-1}(l(m-l){)}^k\frac{1}{m-1}\\ & \sim m^{2k}{\int }_0^1(x(1-x){)}^k\mathrm{d}x=\frac{(k!{)}^2}{(2k+1)!}{m}^{2k}(m\to \infty ),\end{array}$$

$$X_m=\frac{2}{3}\frac{{\tilde{X}}_m}{{\mu }_m}=4\frac{{\tilde{X}}_m}{m^2}.$$

$$M(t)=\sum _{k=0}^{\infty }\frac{\mathbb{E}[X^k]}{k!}(-t{)}^k=\frac{1}{2}\sqrt{\frac{\pi }{t}}{e}^t\text{mbox}erf\left(\sqrt{t}\right).$$

$$p(x)=\left\{\begin{array}[]{cr}\frac{1}{2\sqrt{1-x}}&0\leq x\leq 1\\ 0&x>1\end{array}\right..$$

$$P(x,q)=\sum _{m,n}{p}_{m,n}{x}^m{q}^n.$$

$$\mathbb{E}[({\tilde{X}}_m{)}_k]=\frac{{\sum }_n(n{)}_k{p}_{m,n}}{{\sum }_n{p}_{m,n}}=\frac{[x^m]{\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}}{[x^m]P(x,1)},$$

$$P(x,q)=x^{2}qP(qx,q)+\frac{x^{2}q(1+qx)}{1-qx}.$$

$$g_k(x)=\frac{1}{k!}{\frac{{\partial }^k}{\partial q^k}P(x,q)|}_{q=1}$$

$$\begin{array}{rl}{g}_0(x)& =\frac{x^2}{(1-x{)}^2},g_1(x)=\frac{x^2}{(1-x{)}^4},\\ g_2(x)& =\frac{2x^3}{(1-x{)}^6},g_3(x)=\frac{6x^4}{(1-x{)}^8},\\ g_4(x)& =\frac{x^4(1+22x+x^2)}{(1-x{)}^{10}},g_5(x)=\frac{12x^5(1+8x+x^2)}{(1-x{)}^{12}}.\end{array}$$

$$g_k(x)\sim \frac{k!}{(1-x{)}^{2k+2}}(x\to 1).$$

$$\frac{\mathbb{E}[(X_m{)}^k]}{k!}\sim \frac{\mathbb{E}[(X_m{)}_k]}{k!}\sim \frac{k!}{\Gamma (2k+2)}=\frac{k!}{(2k+1)!}(m\to \infty ),$$

Full text

1 Introduction

For a given combinatorial class of objects, such as polygons or polyhedra, the most basic question concerns the number of objects of a given size (always assumed to be finite), or an asymptotic estimate thereof. Informally stated, in this overview we will analyse the refined question:

What does a typical object look like?

In contrast to the combinatorial question about the number of objects of a given size, the latter question is of a probabilistic nature. For counting parameters in addition to object size, one asks for their (asymptotic) probability law. To give this question a meaning, an underlying ensemble has to be specified. The simplest choice is the uniform ensemble, where each object of a given size occurs with equal probability.

For self-avoiding polygons on the square lattice, size may be the number of edges of the polygon, and an additional counting parameter may be the area enclosed by the polygon. We will call this ensemble the fixed perimeter ensemble. For the uniform fixed perimeter ensemble, one assumes that, for a fixed number of edges, each polygon occurs with the same probability. Another ensemble, which we will call the fixed area ensemble, is obtained with size being the polygon area, and the number of edges being an additional counting parameter. For the uniform fixed area ensemble, one assumes that, for fixed area, each polygon occurs with the same probability.

To be specific, let $p_{m,n}$ denote the number of square lattice self-avoiding polygons of half-perimeter $m$ and area $n$. Discrete random variables $\widetilde{X}_{m}$ of area in the uniform fixed perimeter ensemble and of perimeter $\widetilde{Y}_{n}$ in the uniform fixed area ensemble are defined by

[TABLE]

We are interested in an asymptotic description of these probability laws, in the limit of infinite object size.

In statistical physics, certain non-uniform ensembles are important. For fixed object size, the probability of an object with value $n$ of the counting parameter (such as the area of a polygon) may be proportional to $a^{n}$, for some non-negative parameter $a=\operatorname{e}^{-\beta E}$ of non-uniformity. Here $E$ is the energy of the object, and $\beta=1/(k_{B}T)$, where $T$ is the temperature, and $k_{B}$ denotes Boltzmann’s constant. A qualitative change in the behaviour of typical objects may then be reflected in a qualitative change in the probability law of the counting parameter w.r.t. $a$. Such a change is an indication of a phase transition, i.e., a non-analyticity in the free energy of the corresponding ensemble.

For self-avoiding polygons in the fixed perimeter ensemble, let $q$ denote the parameter of non-uniformity,

[TABLE]

Polygons of large area are suppressed in probability for small values of $q$, such that one expects a typical self-avoiding polygon to closely resemble a branched polymer. Likewise, for large values of $q$, a typical polygon is expected to be inflated, closely resembling a ball (or square) shape. Let us define the ball-shaped phase by the condition that the mean area of a polygon grows quadratically with its perimeter. The ball-shaped phase occurs for $q>1$ [31]. Linear growth of the mean area w.r.t. perimeter is expected to occur for all values $0<q<1$. This phase called the branched polymer phase. Of particular interest is the point $q=1$, at which a phase transition occurs [31]. This transition is called a collapse transition. Similar considerations apply for self-avoiding polygons in the fixed area ensemble,

[TABLE]

with parameter of non-uniformity $x$, where $0<x<\infty$.

For a given model, these effects may be studied using data from exact or Monte-Carlo enumeration and series extrapolation techniques. Sometimes, the underlying model is exactly solvable, i.e., it obeys a combinatorial decomposition, which leads to a recursion for the counting parameter. In that case, its (asymptotic) behaviour may be extracted from the recurrence.

A convenient tool is generating functions. The combinatorial information about the number of objects of a given size is coded in a one-variable (ordinary) generating function, typically of positive and finite radius of convergence. Given the generating function of the counting problem, the asymptotic behaviour of its coefficients can be inferred from the leading singular behaviour of the generating function. This is determined by the location and nature of the singularity of the generating function closest to the origin. There are elaborate techniques for studying this behaviour exactly [37] or numerically [43].

The case of additional counting parameters leads to a multivariate generating function. For self-avoiding polygons, the half-perimeter and area generating function is

[TABLE]

For a fixed value of a non-uniformity parameter $q_{0}$, where $0<q_{0}\leq 1$, let $x_{0}$ be the radius of convergence of $P(x,q_{0})$. The asymptotic law of the counting parameter is encoded in the singular behaviour of the generating function $P(x,q)$ about $(x_{0},q_{0})$. If locally about $(x_{0},q_{0})$ the nature of the singularity of $P(x,q)$ does not change, then distributions are expected to be concentrated, with a Gaussian limit law. This corresponds to the physical intuition that fluctuations of macroscopic quantities are asymptotically negligible away from phase transition points. If the nature of the singularity does change locally, we expect non-concentrated distributions, resulting in non-Gaussian limit laws. This is expected to be the case at phase transition points.

Qualitative information about the singularity structure is given by the singularity diagram (also called the phase diagram). It displays the region of convergence of the two-variable generating function, i.e., the set of points $(x,q)$ in the closed upper right quadrant of the plane, such that the generating function $P(x,q)$ converges. The set of boundary points with positive coordinates is a set of singular points of $P(x,q)$, called the critical curve. See Figure 1 for a sketch of the singularity diagram of a typical polygon model such as self-avoiding polygons, counted by half-perimeter and area, with generating function $P(x,q)$ as above.

There appear two lines of singularities, which intersect at the point $(x,q)=(x_{c},1)$. Here $x_{c}$ is the radius of convergence of the half-perimeter generating function $P(x,1)$, also called the critical point. The nature of a singularity does not change along each of the two lines, and the intersection point $(x,q)=(x_{c},1)$ of the two lines is a phase transition point. For $0<q<1$ fixed, denote by $x_{c}(q)$ the radius of convergence of $P(x,q)$. The branched polymer phase for the fixed perimeter ensemble $0<q<1$ (and also for the corresponding fixed area ensemble) is asymptotically described by the singularity of $P(x,q)$ about $(x_{c}(q),q)$. In the ball-shaped phase $q>1$ of the fixed perimeter ensemble, the (ordinary) generating function does not seem the right object to study, since it has zero radius of convergence for fixed $q>1$. The singularity of $P(x,q)$ about $(x,1)$ describes, for $0<x<x_{c}$, a ball-shaped phase in the fixed area ensemble, with a finite average size of a ball.

For points $(x,q)$ within the region of convergence, both $x$ and $y$ positive, the generating function $P(x,q)$ is finite and positive. Thus, such points may be interpreted as parameters in a mixed infinite ensemble

[TABLE]

The limiting law of the counting parameter in the fixed area or fixed perimeter ensemble can be extracted from the leading singular behaviour of the two-variable generating function. There are two different approaches to the problem. The first one consists in analysing, for fixed non-uniformity parameter $a$, the singular behaviour of the remaining one-parameter generating function and its derivatives w.r.t. $a$. This method is also called the method of moments. It can be successfully applied in the fixed perimeter ensemble at the phase transition point. Typically, this results in non-concentrated distributions.

The second approach derives an asymptotic approximation of the two-variable generating function. Away from a phase transition point, such an approximation can be obtained for some classes of models, typically resulting in concentrated distributions, with a Gaussian law for the centred and normalised random variable. However, it is usually difficult to extract such information at a phase transition point. The theory of tricritical scaling seeks to fill this gap, by suggesting and justifying a particular ansatz for an approximation using scaling functions. Knowledge of the approximation may imply knowledge of the quantities analysed in the first approach.

In the following, we give an overview of these two approaches. For the first approach, summarised by the title limit distributions, there are a number of rigorous results, which we will discuss. The second approach, summarised by the title scaling functions, is less developed. For that reason, our presentation will be more descriptive, stating important open questions. We will stress connections between the two approaches, thereby providing a probabilistic interpretation of scaling functions in terms of limit distributions.

2 Polygon models and generating functions

Models of polygons, polyominoes or polyhedra have been studied intensively on the square and cubic lattices. It is believed that the leading asymptotic behaviour of such models, such as the type of limit distribution or critical exponents, is independent of the underlying lattice.

In two dimensions, a number of models of square lattice polygons have been enumerated according to perimeter and area and other parameters, see [7] for a review of models with an exact solution. The majority of such models has an algebraic perimeter generating function. We mention prudent polygons [96, 22, 8] as a notable exception. Of particular importance for polygon models is the fixed perimeter ensemble, since it models two-dimensional vesicle collapse. Another important ensemble is the fixed area ensemble, which serves as a model of ring polymers. The fixed area ensemble may also describe percolation and cluster growth. For example, staircase polygons are models of directed compact percolation [26, 28, 29, 27, 12, 57]. This may be compared to the exactly solvable case of percolation on a tree [42]. The model of self-avoiding polygons is conjectured to describe the hull of critical percolation clusters [60].

In addition to perimeter, other counting parameters have been studied, such as width and height, generalisations of area [89], radius of gyration [53, 64], number of nearest-neighbour interactions [4], last column height [7], and site perimeter [20, 11]. Also, motivated by applications in chemistry, symmetry subclasses of polygon models have been analysed [63, 62, 40, 95]. Whereas this gives rise to a number of different ensembles, only a few of them have been asymptotically studied. Not all of them display phase transitions.

In three dimensions, models of polyhedra on the cubic lattice have been enumerated according to perimeter, surface area and volume, see [74, 102, 3] and the discussion in section 3.9. Various ensembles may be defined, such as the fixed surface area ensemble and the fixed volume ensemble. The fixed surface area ensemble serves as a model of three-dimensional vesicle collapse [104].

In this chapter, we will consider models of square lattice polygons, counted by half-perimeter and area. Let $p_{m,n}$ denote the (finite) number of such polygons of half-perimeter $m$ and area $n$. The numbers $p_{m,n}$ will always satisfy the following assumption.

Assumption 1**.**

For $m,n\in\mathbb{N}_{0}$, let non-negative integers $p_{m,n}\in\mathbb{N}_{0}$ be given. The numbers $p_{m,n}$ are assumed to satisfy the following properties.

• i)

There exist positive constants $A,B>0$ such that $p_{m,n}=0$ if $n\leq Am$ or if $n\geq Bm^{2}$.

• ii)

The sequence $\left(\sum_{n}p_{m,n}\right)_{m\in\mathbb{N}_{0}}$ has infinitely many positive elements and grows at most exponentially.

Remarks. i) A sequence $(a_{n})_{n\in\mathbb{N}_{0}}$ is said to grow at most exponentially, if there are positive constants $C$, $\mu$ such that $|a_{n}|\leq C\mu^{n}$ for all $n$.

ii) Condition $i)$ reflects the geometric constraint that the area of a polygon grows at most quadratically and at least linearly with its perimeter. For self-avoiding polygons, we have $n\geq m-1$. Since $p_{m,n}=0$ if $m<2$, we may choose $A=1/3$. Since $n\leq m^{2}/4$ for self-avoiding polygons, we may choose $B=1/3$. Condition ii) is a natural condition on the growth of the number of polygons of a given perimeter. For self-avoiding polygons, we may choose $C=1$ and $\mu=16$.

iii) For models with counting parameters different from area, or for models in higher dimensions, a modified assumption holds, with the growth condition i) being replaced by $n\leq Am^{k_{0}}$ and $n\geq Bm^{k_{1}}$, for appropriate values of $k_{0}$ and $k_{1}$. Counting parameters statisfying $p_{m,n}=0$ for $n\geq Bm^{k}$ are called rank $k$ parameters [25].

The above assumption imposes restrictions on the generating function of the numbers $p_{m,n}$. These explain the qualitative form of the singularity diagram Figure 1.

Proposition 1**.**

For numbers $p_{m,n}$, let Assumption 1 be satisfied. Then, the generating function $P(x,q)=\sum_{m,n}p_{m,n}x^{m}q^{n}$ has the following properties.

• i)

The generating function $P(x,q)$ satisfies for $k\in\mathbb{N}$

[TABLE]

where $\ll$ denotes coefficient-wise domination.

• ii)

The evaluation $P(x,1)$ is a power series with radius of convergence $x_{c}$, where $0<x_{c}\leq 1$.

• iii)

The generating function $P(x,q)$ diverges, if $x\neq 0$ and $|q|>1$. It converges, if $|q|<1$ and $|x|<x_{c}q^{-A}$. In particular, for $k\in\mathbb{N}_{0}$, the evaluations

[TABLE]

are power series with radius of convergence 1.

• iv)

For $k\in\mathbb{N}_{0}$, the evaluations

[TABLE]

are power series with radius of convergence $x_{c}$. They satisfy, for $|x|<x_{c}$,

[TABLE]

sketch.

The domination formula follows immediately from condition i). The existence of the evaluations at $q=1$ and $x=x_{c}$ as formal power series also follows from condition i). Condition ii) ensures that $0<x_{c}\leq 1$ for the radius of convergence of $P(x,1)$. Equality of the radii of convergence for the derivatives follows from condition i) by elementary estimates. The claimed analytic properties of $P(x,q)$ follow from conditions i) and ii) by elementary estimates. The claimed left-continuity of the derivatives in iv) is implied by Abel’s continuity theorem for real power series. ∎∎

Remarks. i) Proposition 1 implies that the critical curve $x_{c}(q)$ satisfies for $0<q<1$ the estimate $x_{c}(q)\geq x_{c}q^{-A}$. For self-avoiding polygons, the critical curve $x_{c}(q)$ is continuous for $0<q<1$. This follows from a certain supermultiplicative inequality for the numbers $p_{m,n}$ by convexity arguments [48].

ii) Of central importance in the sequel will be the power series

[TABLE]

They are called factorial moment generating functions, for reasons which will become clear later.

We continue studying analytic properties of the factorial moment generating functions. In the following, the notation $x\nearrow x_{0}$ denotes the limit $x\to x_{0}$ for sequences $(x_{n})$ satisfying $|x_{n}|<x_{0}$. The notation $f(x)\sim g(x)$ as $x\nearrow x_{0}$ means that $g(x)\neq 0$ in a left neigbourhood of $x_{0}$ and that $\lim_{x\nearrow x_{0}}f(x)/g(x)=1$. Likewise, $a_{m}\sim b_{m}$ as $m\to\infty$ for sequences $(a_{m}),(b_{m})$ means that $b_{m}\neq 0$ for almost all $m$ and $\lim_{m\to\infty}a_{m}/b_{m}=1$. The following lemma is a standard result.

Lemma 1**.**

Let $(a_{m})_{m\in\mathbb{N}_{0}}$ be a sequence of real numbers, which asymptotically satisfy

[TABLE]

for real numbers $A,x_{c},\gamma$, where $A\neq 0$ and $x_{c}>0$.

Then, the generating function $g(x)=\sum_{m=0}^{\infty}a_{m}x^{m}$ has radius of convergence $x_{c}$. If $\gamma\notin\{0,-1,-2,\ldots\}$, then there exists a power series $g^{(reg)}(x)$ with radius of convergence strictly larger than $x_{c}$, such that $g(x)$ satisfies

[TABLE]

where $\Gamma(z)$ denotes the Gamma function. ∎

Remarks. i) The above lemma can be proved using the analytic properties of the polylog function [32]. If $\gamma\in\{0,-1,-2,\ldots\}$, an asymptotic form similar to Eq. (3) is valid, which involves logarithms.

ii) The function $g^{(reg)}(x)$ in the above lemma is not unique. For example, if $\gamma>0$, any polynomial in $x$ may be chosen. We demand $g^{(reg)}(x)\equiv 0$ in that case. If $\gamma<0$ and $g^{(reg)}(x)$ is restricted to be a polynomial, it is uniquely defined. If $-1<\gamma<0$, we have $g^{(reg)}(x)\equiv g(x_{c})$. In the general case, the polynomial has degree $\lfloor-\gamma\rfloor$, compare [32]. In the following, we will demand uniqueness by the above choice. The power series $g^{(sing)}(x):=\left(g(x)-g^{(reg)}(x)\right)$ is then called the singular part of $g(x)$.

Conversely, let a power series $g(x)$ with radius of convergence $x_{c}$ be given. In order to conclude from Eq. (3) the behaviour Eq. (2), certain additional analyticity assumptions on $g(x)$ have to be satisfied. To this end, a function $g(x)$ is called $\Delta(x_{c},\eta,\phi)$-regular (or simply $\Delta$-regular) [30], if there is a positive real number $x_{c}>0$, such that $g(x)$ is analytic in the indented disc $\Delta(x_{c},\eta,\phi):=\{z\in\mathbb{C}:|z|\leq x_{c}+\eta,|\operatorname{Arg}(z-x_{c})|\geq\phi\}$, for some $\eta>0$ and some $\phi$, where $0<\phi<\pi/2$. Note that $x_{c}\notin\Delta$, where we adopt the convention $\operatorname{Arg}(0)=0$. The point $x=x_{c}$ is the only point for $|x|\leq x_{c}$, where $g(x)$ may possess a singularity.

Lemma 2** ([35]).**

Let the function $g(x)$ be $\Delta$-regular and assume that

[TABLE]

If $\gamma\notin\{0,-1,-2,\ldots\}$, we then have

[TABLE]

where $[x^{m}]g(x)$ denotes the Taylor coefficient of $g(x)$ of order $m$ about $x=0$. ∎

Remarks. i) Note that the coefficients of the function $f(x)=(1-x/x_{c})^{-\gamma}$ with real exponent $\gamma\notin\{0,-1,-2,\ldots\}$ satisfy

[TABLE]

This may be seen by an application of the binomial series and Stirling’s formula. For functions $g(x)\sim f(x)$, the assumption of $\Delta$-regularity for $g(x)$ ensures that the same asymptotic estimate holds for the coefficients of $g(x)$.

ii) Theorems of the above type are called transfer theorems [35, 37]. The set of $\Delta$-regular functions with singularities of the above form is closed under addition, multiplication, differentiation, and integration [30].

iii) The case of a finite number of singularities on the circle of convergence can be treated by a straightforward extension of the above result [35, 37].

Lemma 1 implies a particular singular behaviour of the factorial moment generating functions, if the numbers $p_{m,n}$ satisfy certain typical asymptotic estimates. We write $(a)_{k}=a\cdot(a-1)\cdot\ldots\cdot(a-k+1)$ to denote the lower factorial.

Proposition 2**.**

For $m,n\in\mathbb{N}_{0}$, let real numbers $p_{m,n}$ be given. Assume that the numbers $p_{m,n}$ asymptotically satisfy, for $k\in\mathbb{N}_{0}$,

[TABLE]

for real numbers $A_{k},x_{c},\gamma_{k}$, where $A_{k}>0$, $x_{c}>0$, and $\gamma_{k}\notin\{0,-1,-2,\ldots\}$.

Then, the factorial moment generating functions $g_{k}(x)$ satisfy

[TABLE]

where $f_{k}=A_{k}\,\Gamma(\gamma_{k})$. ∎

Remarks. i) The above assumption on the growth of the coefficients in Eq. (5) is typical for polygon models, with $\gamma_{k}=(k-\theta)/\phi$, and $\phi>0$.

ii) If the numbers $p_{m,n}$ satisfy, in addition to Eq. (5), Condition $i)$ of Assumption 1, this implies for exponents of the form $\gamma_{k}=(k-\theta)/\phi$, where $\phi>0$, the estimate $1/2\leq\phi\leq 1$.

iii) The proposition implies that the singular part of the factorial moment generating function $g_{k}(x)$ is asymptotically equal to the singular part of the corresponding (ordinary) moment generating function,

[TABLE]

We give a list of exponents and area limit distributions for a number of polygon models. An asterisk denotes that corresponding results rely on a numerical analysis. It appears that the value $(\theta,\phi)=(1/3,2/3)$ arises for a large number of models. Furthermore, the exponent $\gamma_{0}$ seems to determine the area limit law. These two observations will be explained in the following section.

3 Limit distributions

In this section, we will concentrate on models of square lattice polygons in the fixed perimeter ensemble, and analyse their area law. The uniform ensemble is of particular interest, since non-Gaussian limit laws usually appear, due to expected phase transitions at $q=1$. For non-uniform ensembles $q\neq 1$, Gaussian limit laws are expected, due to the absence of phase transitions.

There are effective techniques for the uniform ensemble, since the relevant generating functions are typically algebraic. This is different from the fixed area ensemble, where singularities are more difficult to analyse. It will turn out that the dominant singularity of the perimeter generating function determines the limiting area law of the model. We will first discuss several examples with different type of singularity. Then, we will describe a general result, by analysing classes of $q$-difference equations (see e.g. [103]), which exactly solvable polygon models obey. Whereas in the case $q\neq 1$ their theory is developed to some extent, the case $q=1$ is more difficult to analyse. Motivated by the typical behaviour of polygon models, we assume that a $q$-difference equation reduces to an algebraic equation as $q$ approaches unity, and then analyse the behaviour of its solution about $q=1$.

Useful background concerning a probabilistic analysis of counting parameters of combinatorial structures can be found in [37, Ch IX]. See [80, Ch 1] and [5, Ch 1] for background about asymptotic expansions. For properties of formal power series, see [39, Ch 1.1]. A useful reference on the Laplace transform, which will appear below, is [23].

3.1 An illustrative example: Rectangles

3.1.1 Limit law of area

Let $p_{m,n}$ denote the number of rectangles of half-perimeter $m$ and area $n$. Consider the uniform fixed perimeter ensemble, with a discrete random variable of area $\widetilde{X}_{m}$ defined by

[TABLE]

The $k$-th moments of $\widetilde{X}_{m}$ are given explicitly by

[TABLE]

where we approximated the Riemann sum by an integral, using the Euler-MacLaurin summation formula. Thus, the random variable $\widetilde{X}_{m}$ has mean $\mu_{m}\sim m^{2}/6$ and variance $\sigma_{m}^{2}\sim m^{4}/180$. Since the sequence of random variables $(\widetilde{X}_{m})$ does not satisfy the concentration property $\lim_{m\to\infty}\sigma_{m}/\mu_{m}=0$, we expect a non-trivial limiting distribution. Consider the normalised random variable

[TABLE]

Since the moments of $X_{m}$ converge as $m\to\infty$, and the limit sequence $M_{k}:=\lim_{m_{\to}\infty}\mathbb{E}[X_{m}^{k}]$ satisfies the Carleman condition $\sum_{k}(M_{2k})^{-1/(2k)}=\infty$, they define [17, Ch 4.5] a unique random variable $X$ with moments $M_{k}$. Its moment generating function $M(t)=\mathbb{E}[\operatorname{e}^{-tX}]$ is readily obtained as

[TABLE]

The corresponding probability distribution $p(x)$ is obtained by an inverse Laplace transform, and is given by

[TABLE]

This distribution is known as the beta distribution $\beta_{1,1/2}$. Together with [17, Thm 4.5.5], we arrive at the following result.

Theorem 1**.**

The area random variable $\widetilde{X}_{m}$ of rectangles Eq. (7) has mean $\mu_{m}\sim m^{2}/6$ and variance $\sigma_{m}^{2}\sim m^{4}/180$. The normalised random variables $X_{m}$ Eq. (8) converge in distribution to a continuous random variable with limit law $\beta_{1,1/2}$ Eq. (9). We also have moment convergence. ∎

3.1.2 Limit law via generating functions

We now extract the limit distribution using generating functions. Whereas the derivation is less direct than the previous approach, the method applies to a number of other cases, where a direct approach fails. Consider the half-perimeter and area generating function $P(x,q)$ for rectangles,

[TABLE]

The factorial moments of the area random variable $\widetilde{X}_{m}$ Eq. (7) are obtained from the generating function via

[TABLE]

where $(a)_{k}=a\cdot(a-1)\cdot\ldots\cdot(a-k+1)$ is the lower factorial. The generating function $P(x,q)$ satisfies [87, Eq. 5.1] the linear $q$-difference equation [103]

[TABLE]

Due to the particular structure of the functional equation, the area moment generating functions

[TABLE]

are rational functions and can be computed recursively from the functional equation, by repeated differentiation w.r.t. $q$ and then setting $q$=1. (Such calculations are easily performed with a computer algebra system.) This gives, in particular,

[TABLE]

Whereas the exact expressions get messy for increasing $k$, their asymptotic form about their singularity $x_{c}=1$ is simply given by

[TABLE]

The above result can be inferred from the functional equation, which induces a recursion for the functions $g_{k}(x)$, which in turn can be asymptotically analysed. This method is called moment pumping [36]. Below, we will extract the above asymptotic behaviour by the method of dominant balance.

The asymptotic behaviour of the moments of $\widetilde{X}_{m}$ can be obtained from singularity analysis of generating functions, as described in Lemma 2. Using the functional equation, it can be shown that all functions $g_{k}(x)$ are Laurent series about $x=1$, with a finite number of terms. Hence the remark following Lemma 2 implies for the (factorial) moments of the random variable $X_{m}$ Eq. (8) the expression

[TABLE]

in accordance with the previous derivation.

On the level of the moment generating function, an application of Watson’s lemma [5, Sec 4.1] shows that the coefficients $k!$ in Eq. (11) appear in the asymptotic expansion of a certain Laplace transform of the (entire) moment generating function $\mathbb{E}[\operatorname{e}^{-tX}]$,

[TABLE]

Note that the r.h.s. is formally obtained by term-by-term integration of the l.h.s..

Using the arguments of [46, Ch 8.11], one concludes that there exists an $s_{0}>0$, such that there is a unique function $F(s)$ analytic for $\Re(s)\geq s_{0}$ with the above asymptotic expansion. It is given by

[TABLE]

where $\operatorname{Ei}(z)=\int_{1}^{\infty}\operatorname{e}^{-tz}/t\,{\rm d}t$ is the exponential integral. The moment generating function $M(t)=\mathbb{E}[\operatorname{e}^{-tX}]$ of the random variable $X$ is given by an inverse Laplace transform of $F(s)$,

[TABLE]

Since there are effective methods for computing inverse Laplace transforms [23], the question arises whether the function $F(s)$ can be easily obtained. It turns out that the functional equation Eq. (10) induces a differential equation for $F(s)$. This equation can be obtained in a mechanical way, using the method of dominant balance.

3.1.3 Dominant balance

For a given functional equation, the method of dominant balance consists of a certain rescaling of the variables, such that the quantity of interest appears in the expansion of a rescaled variable to leading order. The method was originally used as an heuristic tool in order to extract the scaling function of a polygon model [84] (see the following section). In the present framework, it is a rigorous method.

Consider the half-perimeter and area generating function $P(x,q)$ as a formal power series. The substitution $q=1-\widetilde{\epsilon}$ is valid, since the coefficients of the power series $P(x,q)$ in $x$ are polynomials in $q$. We get the power series in $\widetilde{\epsilon}$,

[TABLE]

whose coefficients $(-1)^{k}g_{k}(x)$ are power series in $x$. The functional equation Eq. (10) induces an equation for $H(x,\widetilde{\epsilon})$, from which the factorial area moment generating functions $g_{k}(x)$ may be computed recursively.

Now replace $g_{k}(x)$ by its expansion about $x=1$,

[TABLE]

Introducing $\widetilde{s}=1-x$, this leads to a power series $E(\widetilde{s},\widetilde{\epsilon})$ in $\widetilde{\epsilon}$,

[TABLE]

whose coefficients are Laurent series in $\widetilde{s}$. As above, the functional equation induces an equation for the power series $E(\widetilde{s},\widetilde{\epsilon})$ in $\widetilde{\epsilon}$, from which the expansion coefficients may be computed recursively.

We infer from the previous equation that

[TABLE]

Write $F(s,\epsilon)=\sum_{l\geq 0}F_{l}(s)\epsilon^{l}$. By construction, the (formal) series $F_{0}(s)=F(s,0)$ coincides with the asymptotic expansion of the desired function $F(s)$ Eq. (12) about infinity.

The above example suggests a technique for computing $F_{0}(s)$. The functional equation Eq. (10) for $P(x,q)$ induces, after reparametrisation, differential equations for the functions $F_{l}(s)$, from which $F_{0}(s)$ may be obtained explicitly. These may be computed by first writing

[TABLE]

and then introducing variables $s$ and $\epsilon$, by setting $x=1-s\epsilon$ and $q=1-\epsilon^{2}$. Expand the equation to leading order in $\epsilon$. This yields, to order $\epsilon^{0}$, the first order differential equation

[TABLE]

The above equation translates into a recursion for the coefficients $f_{k,0}$, from which $f_{k,0}=k!$ can be deduced. In addition, the equation has a unique solution with the prescribed asymptotic behaviour Eqn. (13), which is given by $F_{0}(s)=\operatorname{Ei}(s^{2})\operatorname{e}^{s^{2}}$.

As we will argue in the next section, Eq. (14) is sometimes referred to as a scaling Ansatz, the function $F(s,0)$ appears as a scaling function, the functions $F_{l}(s)$, for $l\geq 1$, appear as correction-to-scaling functions. In our formal framework, where the series $F_{l}(s)$ are rescaled generating functions for the coefficients $f_{k,l}$, their derivation is rigorous.

3.2 A general method

In the preceding two subsections, we described a method for obtaining limit laws of counting parameters, via a generating function approach. Since this method will be important in the remainder of this section, we summarise it here. Its first ingredient is based on the so-called method of moments [17, Thm 4.5.5].

Proposition 3**.**

For $m,n\in\mathbb{N}_{0}$, let real numbers $p_{m,n}$ be given. Assume that the numbers $p_{m,n}$ asymptotically satisfy, for $k\in\mathbb{N}_{0}$,

[TABLE]

where $A_{k}$ are positive numbers, and $\gamma_{k}=(k-\theta)/\phi$, with real constants $\theta$ and $\phi>0$. Assume that the numbers $M_{k}:=A_{k}/A_{0}$ satisfy the Carleman condition

[TABLE]

Then the following conclusions hold.

• i)

For almost all $m$, the random variables $\widetilde{X}_{m}$

[TABLE]

are well defined. We have

[TABLE]

for a unique random variable $X$ with moments $M_{k}$, where $\stackrel{{\scriptstyle d}}{{}}$ denotes convergence in distribution. We also have moment convergence.

• ii)

If the numbers $M_{k}$ satisfy for all $t\in\mathbb{R}$ the estimate

[TABLE]

then the moment generating function $M(t)=\mathbb{E}[\operatorname{e}^{-tX}]$ of $X$ is an entire function. The coefficients $A_{k}\Gamma(\gamma_{k})$ are related to $M(t)$ by a Laplace transform which has, for $\theta>0$, the asymptotic expansion

[TABLE]

sketch.

A straightforward calculation using Eq. (15) leads to

[TABLE]

This implies that the same asymptotic form holds for the (ordinary) moments $\mathbb{E}[(\widetilde{X}_{m})^{k}]$. Due to the growth condition Eq. (16), the sequence $(M_{k})$ defines a unique random variable $X$ with moments $M_{k}$. Also, moment convergence of the sequence $(X_{m})$ to $X$ implies convergence in distribution, see [17, Thm 4.5.5]. Due to the growth condition Eq. (19), the function $M(t)$ is entire. Hence the conditions of Watson’s Lemma [5, Sec 4.1] are satisfied, and we obtain Eq. (20). ∎∎

Remarks. i) The growth condition Eq. (19) implies the Carleman condition Eq. (16). All examples below have entire moment generating functions $M(t)$.

ii) If $\gamma_{0}<0$, a modified version of Eq. (20) can be given, see for example staircase polygons below.

Proposition 2 states that assumption Eq. (15) translates, at the level of the half-perimeter and area generating function $P(x,q)=\sum_{m,n}p_{m,n}x^{m}q^{n}$, to a certain asymptotic expression for the factorial moment generating functions

[TABLE]

Their asymptotic behaviour follows from Eq. (15), and is

[TABLE]

where $f_{k}=A_{k}\Gamma(\gamma_{k})$. Adopting the generating function viewpoint, the amplitudes $f_{k}$ determine the numbers $A_{k}$, hence the moments $M_{k}=A_{k}/A_{0}$ of the limit distribution. The series $F(s)=\sum_{k\geq 0}(-1)^{k}f_{k}s^{-\gamma_{k}}$ will be of central importance in the sequel.

Definition 1** (Area amplitude series).**

Let Assumption 1 be satisfied. Assume that the generating function $P(x,q)=\sum_{m,n}p_{m,n}x^{m}q^{n}$ satisfies asymptotically

[TABLE]

with exponents $\gamma_{k}\notin\{0,-1,-2,\ldots\}$. Then, the formal series

[TABLE]

is called the area amplitude series.

Remarks. i) Proposition 3 states that the area amplitude series appears in the asymptotic expansion about infinity of a Laplace transform of the moment generating function of the area limit distribution. The probability distribution of the limiting area distribution is related to $F(s)$ by a double Laplace transform.

ii) For typical polygon models, all derivatives of $P(x,q)$ w.r.t. $q$, evaluated at $q=1$, exist and have the same radius of convergence, see Proposition 1. Typical polygon models do have factorial moment generating functions of the above form, see the examples below.

The second ingredient of the method consists in applying the method of dominant balance. As described above, this may result in a differential equation (or in a difference equation [90]) for the function $F(s)$. Its applicability has to be tested for each given type of functional equation. Typically, it can be applied if the factorial area moment generating functions $g_{k}(x)$ Eq. (1) have, for values $x<x_{c}$, a local expansion about $x=x_{c}$ of the form

[TABLE]

where $\gamma_{k,l}=(k-\theta_{l})/\phi$ and $\theta_{l+1}>\theta_{l}$. If a transfer theorem such as Lemma 2 applies, then the differential equation for $F(s)$ induces a recurrence for the moments of the limit distribution. If the differential equation can be solved in closed form, inverse Laplace transform techniques may be applied in order to obtain explicit expressions for the moment generating function and the probability density. Also, higher order corrections to the limiting behaviour may be analysed, by studying the functions $F_{l}(s)$, for $l\geq 1$. See [87] for examples.

3.3 Further examples

Using the general method as described above, area limit laws for the other exactly solved polygon models can be derived. A model with the same area limit law as rectangles is convex polygons, compare [87]. We will discuss some classes of polygon models with different area limit laws.

3.3.1 Ferrers diagrams

In contrast to the previous example, the limit distribution of area of Ferrers diagrams is concentrated.

Proposition 4**.**

The area random variable $\widetilde{X}_{m}$ of Ferrers diagrams has mean $\mu_{m}\sim m^{2}/8$. The normalised random variables $X_{m}$ Eq. (18) converge in distribution to a random variable with density $p(x)=\delta(x-1/8)$.

Remark. It should be noted that the above convergence statement already follows from the concentration property $\lim_{m\to\infty}\sigma_{m}/\mu_{m}=0$, with $\sigma_{m}^{2}\sim m^{3}/48$ the variance of $X_{m}$, by an explicit analysis of the first three factorial moment generating functions. (By Chebyshev’s inequality, the concentration property implies convergence in probability, which in turn implies convergence in distribution.) For illustrative purposes, we follow a different route via the moment method in the following proof.

Proof.

Ferrers diagrams, counted by half-perimeter and area, satisfy the linear $q$-difference equation [87, Eq (5.4)]

[TABLE]

The perimeter generating function $g_{0}(x)=x^{2}/(1-2x)$ is obtained by setting $q=1$ in the above equation. Hence $x_{c}=1/2$. Using the functional equation, it can be shown by induction on $k$ that all area moment generating functions $g_{k}(x)$ are rational in $g_{0}(x)$ and its derivatives. Hence all $g_{k}(x)$ are rational functions. Since the area of a polygon grows at most quadratically with the perimeter, we have a bound on the exponent, $\gamma_{k}\leq 2k+1$, of the leading singular part of $g_{k}(x)$. Given this bound, the method of dominant balance can be applied. We set

[TABLE]

and introduce new variables $s$ and $\epsilon$ by $q=1-\epsilon^{2}$ and $2x=1-s\epsilon$. Then an expansion of the functional equation yields, to order $\epsilon^{0}$, the ODE of first order $F^{\prime}(s)=4sF(s)-1$, whose unique solution with the prescribed asymptotic behaviour is

[TABLE]

It can be inferred from the differential equation that all coefficients in the asymptotic expansion of $F(s)$ at infinity are nonzero. Hence, the above exponent bound is tight. It can be inferred from the functional equation by induction on $k$ that each $g_{k}(x)$ is a Laurent polynomial about $x_{c}=1/2$. Thus, Lemma 2 applies, and we obtain the moment generating function of the corresponding random variable Eq. (18) as $M(s)=\exp(-s/8)$. This is readily recognised as the moment generating function of a probability distribution concentrated at $x=1/8$. ∎∎

A sequence of random variables, which satisfies the concentration property, often leads to a Gaussian limit law, after centering and suitable normalisation. This is also the case for Ferrers diagrams.

Theorem 2** ([97]).**

The area random variable $\widetilde{X}_{m}$ of Ferrers diagrams has mean $\mu_{m}\sim m^{2}/8$ and variance $\sigma_{m}^{2}\sim m^{3}/48$. The centred and normalised random variables

[TABLE]

converge in distribution to a Gaussian random variable. ∎

Remarks. i) It is possible to prove this result by the method of dominant balance. The idea of proof consists in studying the functional equation of the generating function for the “centred coefficients” $p_{m,n}-\mu_{m}$.

ii) The above arguments can also be applied to stack polygons to yield the concentration property and a central limit theorem.

3.3.2 Staircase polygons

The limit law of area of staircase polygons is the Airy distribution. This distribution (see [34] and the survey [52]) is conveniently defined via its moments.

Definition 2** (Airy distribution [34]).**

The random variable $Y$ is said to be Airy distributed if

[TABLE]

where $\gamma_{k}=3k/2-1/2$, and the numbers $\phi_{k}$ satisfy, for $k\geq 1$, the quadratic recurrence

[TABLE]

with initial condition $\phi_{0}=-1$.

Remarks ([34, 58]). i) The first moment is $\mathbb{E}[Y]=\sqrt{\pi}$. The sequence of moments can be shown to satisfy the Carleman condition. Hence the distribution is uniquely determined by its moments.

ii) The numbers $\phi_{k}$ appear in the asymptotic expansion of the logarithmic derivative of the Airy function at infinity,

[TABLE]

where $\operatorname{Ai}(x)=\frac{1}{\pi}\int_{0}^{\infty}\cos(t^{3}/3+tx)\,{\rm d}t$ is the Airy function.

iii) Explicit expressions for the numbers $\phi_{k}$ are known [58]. They are, for $k\geq 1$, given by

[TABLE]

where $\operatorname{Bi}(z)$ is the second standard solution of the Airy differential equation $f^{\prime\prime}(z)-zf(z)=0$.

iv) The Airy distribution appears in a variety of contexts [34]. In particular, the random variable $Y/\sqrt{8}$ describes the law of the area of a Brownian excursion. See also [76] for an overview from a physical perspective.

Explicit expressions have been derived for the moment generating function of the Airy distribution and for its density.

Fact 1** ([19, 66, 99, 34]).**

The moment generating function $M(t)=\mathbb{E}[\operatorname{e}^{-tY}]$ of the Airy distribution satisfies the modified Laplace transform

[TABLE]

The moment generating function $M(t)$ is given explicitly by

[TABLE]

where the numbers $-\beta_{k}$ are the zeros of the Airy function. Its density $p(x)$ is given explicitly by

[TABLE]

where $v_{k}=2\beta_{k}^{3}/(27x^{2})$ and $U(a,b,z)$ is the confluent hypergeometric function.∎

Remarks. i) The confluent hypergeometric function $U(a,b;z)$ is defined as [1]

[TABLE]

where ${}_{1}F_{1}[a;b;z]$ is the hypergeometric function

[TABLE]

ii) The moment generating function and its density are obtained by two consecutive inverse Laplace transforms of Eq. (22), see [67, 68] and [99, 54].

iii) In the proof of the following theorem, we will derive Eq. (22) using the model of staircase polygons. This shows, in particular, that the coefficients $\phi_{k}$ appear in the asymptotic expansion of the Airy function.

Theorem 3**.**

The normalised area random variables $X_{m}$ of staircase polygons Eq. (18) satisfy

[TABLE]

where $Y$ is Airy distributed according to Definition 2. We also have moment convergence.

Remark. Given the functional equation of the half-perimeter and area generating function of staircase polygons,

[TABLE]

(see [88] for a recent derivation), this result is a special case of Theorem 4 below, which is stated in [25].

Proof.

We use the method of dominant balance. From the functional equation Eq. (23), we infer $g_{0}(x)=1/4+\sqrt{1-4x}/2+(1-4x)/4$. Hence $x_{c}=1/4$. The structure of the functional equation implies that all functions $g_{k}(x)$ can be written as Laurent series in $s=\sqrt{1-4x}$, see also Proposition 7 below. Explicitly, we get $g_{1}(x)=x^{2}/(1-4x)$. This suggests $\gamma_{k}=(3k-1)/2$. An upper bound of this form on the exponent $\gamma_{k}$ can be derived without too much effort from the functional equation, by an application of Faa di Bruno’s formula, see also [89, Prop (4.4)]. Thus, the method of dominant balance can be applied. We set

[TABLE]

and introduce variables $s,\epsilon$ by $4x=1-s\epsilon^{2}$ and $q=1-\epsilon^{3}$. In the above equation, we excluded the constant $1/4=:P^{(reg)}(x,q)$, since it does not contribute to the moment asymptotics. Expanding the functional equation to order $\epsilon^{2}$ gives the Riccati equation

[TABLE]

It follows that the coefficients $f_{k}$ of $F(s)$ satisfy, for $k\geq 1$, the quadratic recursion

[TABLE]

with initial condition $f_{0}=-1/2$. A comparison with the definition of the Airy distribution shows that $\phi_{k}=2^{2k+1}f_{k}$. Using the closure properties of $\Delta$-regular functions, it can be inferred from the functional equation that (the analytic continuation of) each factorial moment generating function $g_{k}(x)$ is $\Delta$-regular, with $x_{c}=1/4$, see also Proposition 7 below. Hence the transfer theorem Lemma 2 can be applied. We obtain $4X_{m}\stackrel{{\scriptstyle d}}{{\to}}Y$ in distribution and for moments, where $Y$ is Airy distributed. ∎∎

Remarks. i) The unique solution $F(s)$ of the differential equation in the above proof Eq. (24), satisfying the prescribed asymptotic behaviour, is given by

[TABLE]

The moment generating function $M(t)$ of the limiting random variable $X=\lim_{m\to\infty}X_{m}$ is related to the function $F(s)$ via the modified Laplace transform

[TABLE]

where the modification has been introduced in order to ensure a finite integral about the origin. This result relates the above proof to Proposition 1.

ii) The method of dominant balance can be used to obtain corrections $F_{l}(s)$ to the limiting behaviour [87].

The fact that the area law of staircase polygons is, up to normalisation, the same as that of the area under a Brownian excursion, suggests that there might be a combinatorial explanation. Indeed, as is well known, there is a bijection [21, 98] between staircase polygons and Dyck paths, a discrete version of Brownian excursions [2], see figure 2 [88]. Within this bijection, the polygon area corresponds to the sum of peak heights of the Dyck path, but not to the area below the Dyck path.

For more about this connection, see the remark at the end of the following subsection.

3.4 $q$-difference equations

All polygon models discussed above have an algebraic perimeter generating function. Moreover, their half-perimeter and area generating function satisfies a functional equation of the form

[TABLE]

for a real polynomial $G(x,q,y_{0},y_{1})$. Since, under mild assumptions on $G$, the equation reduces to an algebraic equation for $P(x,1)$ in the limit $q\to 1$, it may be viewed as a “deformation” of an algebraic equation. In this subsection, we will analyse equations of this type at the special point $(x,q)=(x_{c},1)$, where $x_{c}$ is the radius of convergence of $P(x,1)$. It will appear that the methods used in the above examples also can be applied to this more general case.

The above equation falls into the class of $q$-difference equations [103]. While particular examples appear in combinatorics in a number of places, see e.g. [37], the asymptotic behaviour of equations of the above form seems to have been systematically studied initially in [25, 87]. The study can be done in some generality, e.g., also for non-polynomial power series $G$, for replacements more general than $x\mapsto qx$, and for multivariate generalisations, see [89] and [25]. For simplicity, we will concentrate on polynomial $G$, and then briefly discuss generalisations. Our exposition closely follows [89, 87].

3.4.1 Algebraic $q$-difference equations

Definition 3** (Algebraic $q$-difference equation [25, 87]).**

An algebraic $q$-difference equation is an equation of the form

[TABLE]

where $G(x,q,y_{0},y_{1},\ldots,y_{N})$ is a complex polynomial. We require that

[TABLE]

Remarks. i) See [103] for an overview of the theory of $q$-difference equations. As $q$ approaches unity, the above equation reduces to an algebraic equation.

ii) Asymptotics for solutions of algebraic $q$-difference equations have been considered in [25]. The above definition is a special case of [89, Def 2.4], where a multivariate extension is considered, and where $G$ may be non-polynomial. Also, replacements more general than $x\mapsto f(q)x$ are allowed. Such equations are called $q$-functional equations in [89]. The results presented below apply mutatis mutandis also to $q$-functional equations.

The algebraic $q$-difference equation in Definition 3 uniquely defines a (formal) power series $P(x,q)$ satisfying $P(0,q)\equiv 0$. This is shown by analysing the implied recurrence for the coefficients $p_{m}(q)$ of $P(x,q)=\sum_{m>0}p_{m}(q)x^{m}$, see also [89, Prop 2.5]. In fact, $p_{m}(q)$ is a polynomial in $q$. The growth of its degree in $m$ is not larger than $cm^{2}$ for some positive constant $c$, hence the counting parameters are rank 2 parameters [25]. In our situation, such a bound holds, since the area of a polygon grows at most quadratically with its perimeter.

From the preceding discussion, it follows that the factorial moment generating functions

[TABLE]

are well-defined as formal power series. In fact, they can be recursively determined from the $q$-difference equation by implicit differentiation, as a consequence of the following proposition.

Proposition 5** ([87, 89]).**

Consider the derivative of order $k>0$ of an algebraic $q$-difference equation Eq. (26) w.r.t. $q$, evaluated at $q=1$. It is linear in $g_{k}(x)$, and its r.h.s. is a complex polynomial in the power series $g_{l}(x)$ and its derivatives up to order $k-l$, where $l=0,\ldots,k$. ∎

Remarks. i) This statement can be shown by analysing the $k$-th derivative of the $q$-difference equation, using Faa di Bruno’s formula [18].

ii) It follows that every function $g_{k}(x)$ is rational in $g_{l}(x)$ and its derivatives up to order $k-l$, where $0\leq l<k$. Since $G$ is a polynomial, $g_{k}(x)$ is algebraic, by the closure properties of algebraic functions.

We discuss analytic properties of the (analytic continuations of the) factorial moment generating functions $g_{k}(x)$. These are determined by the analytic properties of $g_{0}(x)=P(x,1)$. We discuss the case of a square-root singularity of $P(x,1)$, which often occurs for combinatorial structures, and which is well studied, see e.g. [79, Thm 10.6] or [37, Ch VII.4]. Other cases may be treated similarly. We make the following assumption:

Assumption 2**.**

The $q$-difference equation in Definition 3 has the following properties:

• i)

All coefficients of the polynomial $G(x,q,y_{0},y_{1},\ldots,y_{N})$ are non-negative.

• ii)

The polynomial $Q(x,y):=G(x,1,y,y,\ldots,y)$ satisfies $Q(x,0)\not\equiv 0$ and has degree at least two in $y$.

• iii)

$P(x,1)=\sum_{m\geq 1}p_{m}x^{m}$* is aperiodic, i.e., there exist indices $1\leq i<j<k$ such that $p_{i}p_{j}p_{k}\neq 0$, while $\gcd(j-i,k-i)=1$.*

Remarks. i) The positivity assumption is natural for combinatorial constructions. There are, however, $q$-difference equations with negative coefficients, which arise from systems of $q$-difference equations with non-negative coefficients by reduction. Examples are convex polygons [87, Sec 5.4] and directed convex polygons, see below.

ii) Assumptions $i)$ and $ii)$ result in a square-root singularity as the dominant singularity of $P(x,1)$.

iii) Assumption $iii)$ implies that there is only one singularity of $P(x,1)$ on its circle of convergence. Since $P(x,1)$ has non-negative coefficients only, it occurs on the positive real half-line. The periodic case can be treated by a straightforward extension [37].

An application of the (complex) implicit function theorem ensures that $P(x,1)$ is analytic at the origin. It can be analytically continued, as long as the defining algebraic equation remains invertible. Together with the positivity assumption, one can conclude that there is a number $0<x_{c}<\infty$, such that the analytic continuation of $P(x,1)$ satisfies $y_{c}=\lim_{x\nearrow x_{c}}P(x,1)<\infty$, with

[TABLE]

With the positivity assumption on the coefficients, it follows that

[TABLE]

These conditions characterise the singularity of $P(x,1)$ at $x=x_{c}$ as a square-root. It can be shown that there exists a locally convergent expansion of $P(x,1)$ about $x=x_{c}$, and that $P(x,1)$ is analytic for $|x|<x_{c}$. We have the following result. Recall that a function $f(z)$ is $\Delta$-regular if it is analytic in the indented disc $\Delta=\{z:|z|\leq x_{c}+\eta,|\operatorname{Arg}(z-x_{c})|\geq\phi\}$ for some $\eta>0$ and some $\phi$, where $0<\phi<\pi/2$.

Proposition 6** ([79, 37, 89]).**

Given Assumption 2, the power series $P(x,1)$ is analytic at $x=0$, with radius of convergence $x_{c}$. Its analytic continuation is $\Delta$-regular, with a square-root singularity at $x=x_{c}$ and a local Puiseux expansion

[TABLE]

where $y_{c}=\lim_{x\nearrow x_{c}}P(x,1)<\infty$ and $f_{0,0}=-\sqrt{x_{c}\,C/B}$, for constants $B>0$ and $C>0$ as in Eq. (27). The numbers $f_{0,l}$ can be recursively determined from the $q$-difference equation. ∎

The asymptotic behaviour of $P(x,1)=g_{0}(x)$ carries over to the factorial moment generating functions $g_{k}(x)$.

Proposition 7** ([89]).**

Given Assumption 2, all factorial moment generating functions $g_{k}(x)$ are, for $k\geq 1$, analytic at $x=0$, with radius of convergence $x_{c}$. Their analytic continuations are $\Delta$-regular, with local Puiseux expansions

[TABLE]

where $\gamma_{k}=3k/2-1/2$. The numbers $f_{k,0}=f_{k}$ are, for $k\geq 2$, characterised by the recursion

[TABLE]

and the numbers $f_{0}<0$ and $f_{1}>0$ are given by

[TABLE]

for constants $B>0$ and $C>0$ as in Eq. (27). ∎

Remarks. i) This result can be obtained by a direct analysis of the $q$-difference equation, applying Faa di Bruno’s formula, see also [87, Sec 2.2].

ii) Alternatively, it can be obtained by applying the method of dominant balance to the $q$-difference equation. To this end, one notes that all functions $g_{k}(x)$ are Laurent series in $\sqrt{1-x/x_{c}}$, and that their leading exponents are bounded from above by $\gamma_{k}$. (An upper bound on an exponent is usually easier to obtain than its exact value, since cancellations can be ignored). With these two ingredients, the method of dominant balance, as described above, can be applied. The differential equation of the function $F(s)$ then translates, via a transfer theorem, into the above recursion for the coefficients. See [89, Sec 5].

The above result can be used to infer the limit distribution of area, along the lines of Section 3.2.

Theorem 4** ([25, 89]).**

Let Assumption 2 be satisfied. For the solution of an algebraic $q$-difference equation $P(x,q)=\sum_{m,n}p_{m,n}x^{m}q^{n}$, let $\widetilde{X}_{m}$ denote the random variable

[TABLE]

(which is well-defined for almost all $m$). The mean of $\widetilde{X}_{m}$ is given by

[TABLE]

where the numbers $f_{0}$ and $f_{1}$ are given in Eq. (28). The sequence of normalised random variables $X_{m}$ converges in distribution,

[TABLE]

where $Y$ is Airy distributed according to Definition 2. We also have moment convergence. ∎

Remarks. i) An explicit calculation shows that $\phi_{k}=|f_{0}|^{-1}\left(\frac{|f_{0}|}{2f_{1}}\right)^{k}f_{k}$. Together with Proposition 7, the claim of the proof follows by standard reasoning, as in the examples above.

ii) The above theorem appears in [25, Thm 3.1], together with an indication of the arguments of a proof. [There is a misprint in the definition of $\gamma$ in [25, Thm 3.1]. In our notation $\gamma=4Bf_{1}$.] Within the more general setup of $q$-functional equations, the theorem is a special case of [89, Thm 1.5].

iii) The above theorem is a kind of central limit theorem for combinatorial constructions, since the Airy distribution arises under natural assumptions for a large class of combinatorial constructions. For a connection to certain Brownian motion functionals, see below.

3.4.2 $q$-functional equations and other extensions

We discuss extensions of the above result. Generically, the dominant singularity of $P(x,1)$ is a square-root. The case of a simple pole as dominant singularity, which generalises the example of Ferrers diagrams, has been discussed in [87]. Under weak assumptions, the resulting limit distribution of area is concentrated. Other singularities can also be analysed, as shown in the examples of rectangles above and of directed convex polygons in the following subsection. Compare also [90].

The case of non-polynomial $G$ can be discussed along the same lines, with certain assumptions on the analyticity properties of the series $G$. In the undeformed case $q=1$, it is a classical result [37, Ch VII.3] that the generating function has a square-root as dominant singularity, as in the polynomial case. One can then argue along the above lines that an Airy distribution emerges as the limit law of the deformation variable [89, Thm 1.5]. Such an extension is relevant, since prominent combinatorial models, such as the Cayley tree generating function, fall into that class. See also the discussion of self-avoiding polygons below.

The above statements also remain valid for more general classes of replacements $x\mapsto qx$, e.g., for replacements $x\mapsto f(q)x$, where $f(q)$ is analytic for $0\leq q\leq 1$, with non-negative series coefficients about $q=0$. More interestingly, the idea of introducing a $q$-deformation may be iterated [25], leading to equations such as

[TABLE]

The counting parameters corresponding to $q_{k}$ are rank $k+1$ parameters, and limit distributions for such quantities have been derived for some types of singularities [77, 78, 88]. There is a central limit result for the generic case of a square-root singularity [89]. This generalisation applies to counting parameters, which decompose linearly under a combinatorial construction. These results can also be obtained by an alternative method, which generalises to non-linear parameters, see [51].

The case where the limit $q$ to unity in a $q$-difference equation is not algebraic, has not been discussed. For example, if $G(x,q,P(x,q),P(qx,q))=0$ for some polynomial $G$, the limit $q$ to unity might lead to an algebraic differential equation for $P(x,1)$. This may be seen by noting that

[TABLE]

for $f(x)$ differentiable at $x$. Such equations are possibly related to polygon models such as three-choice polygons [44] or punctured staircase polygons [45]. Their perimeter generating function is not algebraic, hence the models do not satisfy an algebraic $q$-difference equation as in Definition 3.

3.4.3 A stochastic connection

Lastly, we indicate a link to Brownian motion, which appears in [99, 100] and was further developed in [77, 78, 89, 88]. As we saw in Section 3.2, limit distributions can, under certain conditions, be characterised by a certain Laplace transform of their moment generating functions. This approach, which arises naturally from the viewpoint of generating functions, can be applied to discrete versions of Brownian motion, excursions, bridges or meanders. Asymptotic results are results for the corresponding stochastic objects. In fact, distributions of some functionals of Brownian motion have apparently first been obtained using this approach [99, 100].

Interestingly, a similar characterisation appears in stochastics for functionals of Brownian motion, via the Feynman-Kac formula. For example, Louchard’s formula [66] relates the logarithmic derivate of the Airy function to a certain Laplace transform of the moment generating function of the law of the Brownian excursion area. Distributions of functionals of Brownian motion can also be obtained by a path integral approach, see [75] for a recent overview.

The discrete approach provides an alternative method for obtaining information about distributions of certain functionals of Brownian motion. For such functionals, it provides an alternative proof of Louchard’s formula [77, 78]. It leads, via the method of dominant balance, quite directly to moment recurrences for the underlying distribution. These have been studied in the case of rank $$